Understanding Lagrange Multipliers: Solving for Max and Min Values

[Dx]

Find max and min value…f(x,y,z) =3x+2y+z; x2 + y2+z2 = 1

If g(x,y,z) = x2 + y2+z2 = 1 then what do I do next?

I need help to further solve for this please? I am horrible at math and don't understand lagrange multipiers so can anyone better explain it to me and help me solve for difficult problem.

Dx

 

[Hurkyl]

Well, my text has the following box for lagrange multipliers (interesting theorems / procedures are placed in boxes in the text):

To find the maximum and minimum values of f(x, y, z) subject to the constraint g(x, y, z) = k (assuming these extreme values exist):

(a) Find all values of x, y, z, and λ such that:

[nab]f(x, y, z) = λ [nab]g(x, y, z)

and g(x, y, z) = k

(b) Evaluate f at all the points (x, y, z) that arise from step (a). The largest of these values is the maximum value of f; the smallest is the minimum value of f.

I imagine your text has something similar. Do you understand how to start step (a)?

 

[Dx]

Originally posted by Hurkyl
Well, my text has the following box for lagrange multipliers (interesting theorems / procedures are placed in boxes in the text):



I imagine your text has something similar. Do you understand how to start step (a)?

No, I don't understand this what-so-ever. I see one example in my book that wants to find the points of a rectangular hyperbola and its using partial Dx but I don't know how to perform step a could you please help me.

Dx

 

[Hurkyl]

∇f(x, y, z) means the gradient of f with respect to its three variables. The gradient is defined as the row vector:

&nabla;f(x, y, z) = < &part;f(x, y, z)/&part;x, &part;f(x, y, z)/&part;y, &part;f(x, y, z)/&part;z >
I.E. the first coordinate is the partial derivative with respect to (WRT) the first variable, the second coordinate is the partial derivative WRT the second variable, and so on for as many variabes as the function has.


The equation in step (a) is, then, a vector equation, which we solve by setting corresponding coordinates equal:

&nabla;f(x, y, z) = &lambda; &nabla;g(x, y, z)

is

< &part;f(x, y, z)/&part;x, &part;f(x, y, z)/&part;y, &part;f(x, y, z)/&part;z > = &lambda; < &part;g(x, y, z)/&part;x, &part;g(x, y, z)/&part;y, &part;g(x, y, z)/&part;z >

is

&part;f(x, y, z)/&part;x = &lambda; &part;g(x, y, z)/&part;x
&part;f(x, y, z)/&part;y = &lambda; &part;g(x, y, z)/&part;y
&part;f(x, y, z)/&part;z = &lambda; &part;g(x, y, z)/&part;z



(BTW, if you also want me to explain the "why" behind this method, ask and I'll do so... but the "why" may be quite difficult to understand)